Transmission network



May 14, 1935. HW. Bo E 2,001,090

' TRANSMISSION NETWQRK Filed-Sept. 13, 1933 SSheets-Shee't 2 F116. 7 F I68,

FREQUENCY 1':- S INVENTOR HWBODE A TTORNEY May 14, 1935. H w, BOD'E 2,001,090

TRANSMISSION NETWORK Filed Sept. 13, 1955 5 Sheets-SheetB F/GJZ ,4 ie 8 a c I 3 n W W FIG/5 '0 Q E J2 FREQUENCY E0 I REQUENCV 3 1; I a, 17 15 f3 INVENTOR ATTORNEY Patented May 14, 1935 UNITED STATES PATENT oFFics- Hendrik W. Bode, New

York, N. assignor to Bell Telephone Laboratories, Incorporated,

New York, N. Y., a corporation of New York Application September 13, 1933, Serial No. 689,228

8 Claims.

This invention relates to wave filters and more particularly to wave filters of unsymmetrical structure having different image impedances at their input and output terminals.

It has for its principal object to provide in such filters impedance characteristics of the most general possible type subject only to a reciprocal relationship between the impedances at the two Another object is to provide for the conends. trol of the image impedances without affecting the valueof the transfer constant. A further object is to provide an independent control of the image impedancein the case of filters of the series-shunt,ior ladder, type. i

In my earlier Patent No. 1,828,454 issued October 20, 1931, the control of the image impedance and the transfer constant of a symmetrical lattice type wave filter through the allocation of the resonance andanti-resonace frequencies of the branch impedances is described and the stant controls since it permits asmany critical frequencies as may be desired to be introduced into either range independently of the other.

The symmetrical lattice filter provides a very wide range of image impedances and transfer constants but ity is not completely general, that is, capable of producing all possible characteristics, for one reason amongst others, namely,

that itis restricted to networks having the same a image impedances at their input and output terg minals. I have found by mathematical analysis that, so far as the transfer constant is concerned,

" the most general filtercharacteristic is the sum of the transfer constants of a symmetrical and an unsymmetrical network. ,The symmetrical portion can be built as a lattice, the desired transfer constant being obtained by proper allocation of the branch resonances within the transmission band, as explained in my above mentioned patent. The transierconstant of the unsymmetrical portion I have found to correspond to the sum of the transfer constants of known, physically realizable, ladder type half sections. No other types of transfer constant are necessary. a

The most general filter can, thereiore, be constructed bycombining in tandem the lattice network representing the symmetrical portion and the several half sections representing the unsymmetrical portion providing the component structures can be so built that the resulting network will have matched image impedances at its internal junction points. This requires that the unsymmetrical half sections shall be capable of being constructed with at least as wide a range of image impedances as the symmetrical lattice.

It has been pointed out that the image impedance control in the case of the symmetrical lattice is simplified by reason of the fact that the reactances of the line and the lattice branches are everywhere of like sign in the attenuation ranges and are everywhere of opposite sign in the transmission band. In the case of the ladder type of filter, the problem of the image impedance control is complicated by the fact that the transmission and attenuation ranges do not have this simple demarcation. The reactances of the series and shunt branchesare of opposite sign throughout thetransmission band just as in the symmetrical lattice, but in the attenuation ranges the reactances are not everywhereof the same sign and, in certain cases may be of opposite sign at all frequencies. In all cases there exists a range adjacent one or both cut-off frequencies in which the reactances are of the opposite sign and, in the case of the elementary types of filters, for example, a low-pass filter in which the series branches are simple inductances and the shunt branches simple capacities this range may extend over the whole attenuation range.

If it is attempted to control the image impedance of a series-shunt type filter by introducing corresponding resonances or anti-resonances into the branch impedances in the portion of the at tenuation range where the reactances are of opposite sign itvvill be found that this can not be done without changing the ratio of the branch reactances in this range and hence producing a change in the transfer constant of the network. Since the complete control of the image impedance requires that it be capable of assuming zero and infinite values in any frequency range outside the transmission band, the above restriction indicates that the ordinary types of seriesshunt networks can provide only a limited range of impedance characteristics, for a given transfer constant.

The present invention provides networks having the same transfer constants as simple seriesshunt half sections, but having image impedances which are quite general in that their Zeros and poles can be placed anywhere in the attenuation ranges; This generality is obtained by building out the simple prototype with additional series and shunt branches having impedances proportional to the short circuit and open circuit impedancesrespectively of the network to which the branch is added. In this way the ratio of the short circuit and open circuit impedances, and hence the transfer constant, is maintained unchanged.

The nature of the invention will be more fully understood from the following detailed description and the accompanying drawings of which:

Fig. 1 represents in schematic form a general type of network in accordance witlrthe invention;

Fig. 2 shows a prototype of the network of Fig. 1;

Figs. 3 and 4 illustrate respectively a simple band-pass prototype filter and a derived filter in accordance with the invention;

Figs. 5 and 6 show characteristics of the filters of Figs. 3 and 4;

Fig. '7 is a schematic of an alternative form of the network of Fig. 1;

Figs. 8 and 9 represent in schematic form additional general types of the networks of the invention;

Figs. 10 and 11 are explanatory of supplementary methods of impedance control;

Fig. 12 represents a composite filter in accordance with the invention;

Figs. 13 and 14 show prototypes of portions of the filter of Fig. 12; and

Figs. 15 and 16 illustrate the impedance characteristic of the filter of Fig. 12.

The principles of the invention will be readily understood from a consideration of the generalized networks of Figs. 1 and 2 which represent, respectively, one form of the invention and the prototype half section from which it is derived. The transmission parameters of the network of Fig. 1 and its prototype are most readily obtained from the open circuit and short circuit impedances measured at the input and at the output terminals. For the prototype network the open circuit impedance Z10 and the short circuit impedance 210 at terminals I, 2, are respectively and the corresponding impedances Z20 and Z26 at terminals 3, 4, are

Z20 a b The image impedances K1 and K2 at terminals l, 2 and terminals 3, 4 respectively are given by I a I) and Ina/22.22. =w ..zb(1 +1),

where 1' denotes the ratio Za/Zb The transfer constant e of the network is given tanh 0 impedance of the portion to the right. multiplied by the numerical factor (l h) /h. It will be evident that the factor h must be positive and less than unity if the network of Fig. 1 is to be physically realizable. The open and closed circuit impedances 2'10 and Z'Zo of this network measured at terminals l 2, are respectively by At the terminals 3, 4, the corresponding impedances 2'20 and Z'Zc are respectively and (7) From these values the transfer constant, 9, and the image impedances Ki and Kz at terminals I, 2, and terminals 3, 4, respectively, are found to be It will be seen from Equations 8, 9 and 10 that the network has the same transfer constant as that of the prototype but that its image impedances are modified by the introduction of the factor (1+hr), as a multiplier in one case and as a divisor in the other. A change in the image impedances has thus been effected without modification of the transfer or attenuation constant.

The character of the change produced will be illustrated by the application of the transformation to the band-pass prototype filter shown in Fig. 3 of which the series impedance Z9. is made up of inductance L1 and capacity C1 in series and the shunt impedance Zr, comprises inductance L2 and capacity C2 in series. The filter obtained by the transformation described above is shown in Fig. 4, the added series branch comprising two simple resonant combinations connected in parallel.

The curves of Fig. 5 show the impedancefrequency characteristics of the filter of Fig. 3, dotted curves l0 and H representing the reactances of Za and Zb respectively and curve 12 being the image impedance at terminals l, 2. The transmission band extends from frequency ii at which impedance Z2. is resonant to frequency f2 at which Za and Zb are equal and of opposite sign. At frequency is the shunt impedance is resonant and the image impedance has a zero value. This frequency is that of an attenuation peak produced by the shunt resonance. It Will be observed that the two branch reactances have the same sign below f1 and above is, but are of opposite sign in the attenuation range becies.

The values of the image impedances K1 and K2 are given by transformation has the value tweenthe upper cut-off and the peak frequenwhere o denotes 21r times frequency, on andwz corresponds to the cut-off frequencies and (03 to the frequency is of the attenuation peak. The upper cut-off frequency has the value given by corresponding to the conditionthat impedances Za and Zb are equal and of opposite sign.

The frequency factor 1+hr introduced by the where 1 1 h i L2+hL1(CE G) (15) and the image impedances of the transformed network have the values 2 Tech/1 01/ 2 The frequency variations of the new image. impedances are shown bythe curves of Fig. 6 l

of which curve l3 represents the impedance K'1 and dotted curve M, the impedance K'z. In the transmission band where the impedances are real or resistive the effect of the new frequency factors is to produce an additional undulation on each characteristic thus making greater uniformity possible. The new frequency fx at which K1 has a poleand K2 a zero lies in the range between the upper cut-off f2 and the peak frequency f3. That it must lie in this rangefollows from the value given by. equation and from i the fact already shown that the factor It must 4 consisting of hZa/(1h) and Zb/(L-h) in series.

In this case the added shunt branch is equal to the open circuit impedance of the network of Fig. 2, after modificaton of Zb, multiplied by thefactor h/(1-h) An analysis of the network in terms of its open circuit and shortcircuit impedances shows that its image impedances and its transfer constant have the same network of Fig. 1 a

Thederived structures described above have image impedances containing one additional frequency factor, that is, one impedance controlling factor; By continuing the process, additional frequency factorsmay be introduced one at a time without limit. The network, of course, becomes more complex with each subsequent step, but ordinarily it is not necessary for practical purposes to go further than the second step.

values asgiven byequations 8, 9 and 10 for the l The general procedure in obtaining networks' of higher order is illustrated byv Figs. 8 and-9 of which Fig. 8 illustrates the first step in the derivation process and Fig. 9 the general form of the network resulting from two steps of derivation. Fig. 8 represents a network of the type shown in Fig. 1 built out by the addition of a shunt branch adjacent the series branch added by the first derivation. The added shunt branch has animpedance Z1o'/ Z10 being the open circuit impedance of the first derived network meas- .uredat terminals I, 2, and a numerical factor.

The open circuit and short-circuit impedances Z10 and Z10 of the network thus formed have the respective values from which the transfer constant 0" is obtained This expression for the transfer constant'is dif-- ferent from that for the prototype or for the first derivednetwork as given by Equations 4 and "7, but the original value can be regained by giving Z9. and Zb in the network new values such that their new ratio r' satisfies the equation since, obviously the substitution of r in Equation leads to the original transfer constant. This may most readily be accomplished by replacing each Zb impedance by an impedance Zb/h2, such that The resulting network is shown schematically in Fig. 9, the h parameter of the first derived network being designated h1. The image impedances of the network thus obtained have the values and K1." and K2 denoting the image impedances at terminals l, 2, and 3, 4, respectively.

Comparison of Equations 22 with Equations 3 shows that the new image impedances correspond respectively with the image impedances K1 and K2 of the prototype network modified by the introduction of two different frequency factors, one appearing in the numerator and one in the denominator. Each subsequent step of derivation introduces an additional frequency factor, the new factors occurring alternately in the numerator and the denominator.

. The procedure at each stage of derivation may be described as follows: Before adding a new series branch the Z3. impedances throughout the network already developed are modified by being reduced in proportion to the chosen value of h and the new branch is constructed to have an im-- pedance equal to (1-71) /h times the short-circuit impedance of the network to which it is added. Before adding a new, shunt branch the Zb impedances throughout the network are increased inversely in proportion to the chosen value of h and the new branch is made to have an impedance equal to h/ (1--h) times the opencircuitimpedance of the modified network. Inthe design development of a multiple derived network it is desirable that the added branches should not only have the values given above, but should retain the configurations of short-circuit and open-circuit impedances to which they are proportioned. When the network has been fully developed the individual branches may, if necessary, be simplified for constructional purposes by the application of well known theorems relating to equivalent impedances.

In the final networks each series branch retains its proportionality to the short-circuit impedance and each shunt branch retains its proportionality to the open-circuit impedance of the respective portions of the network to which they are added. Consequently, if starting with a single series impedance Za and a single shunt impedance Zb, a network is built out by the addition of alternate series branches proportioned to the successive short-circuit impedances in accordance with proportionality factors h1, h2, hn and shunt branches proportioned as to the successive open circuit impedances in accordance, with factors h1, h2' -hn, the resulting network will have the same transfer constant as a simple prototype consisting of a single series impedance equal to Za multiplied by the product in, H2, -hn and a single short impedance equal to Zb divided by the product hi, 7L2, -hn'.

As in the case of the first derived network, the additional frequency factors introduced by successive derivations correspond to zeros and poles of the image impedances located in the frequency range in which the impedances Za and Zb of the prototype are of opposite signs. By means of the transformations alone it is, therefore, not possible to obtain the most completely general image impedance since this requires that the zeros and. poles of the image impedance can be located anywhere in the attenuation ranges. This problem can however, be solved by a preliminary modification of the prototype network before making the transformation. The required modification consists in substituting for the impedances Za and Zb related impedances which have the same ratio but which are simultaneously resonant or anti-resonant at additional frequencies lying in the ranges where Za and Zb have the same sign. The procedure will be readily understood by the consideration of an illustrative example based on the prototype network ofFig. 3. In that network the branch impedances have the values and (23) their ratio 1 being given by If for Za an impedance Za be substituted having the value .be physically realizable it is necessary that the frequencies corresponding to mo and on be respectively lower than ii the resonance frequency of Za and higher than f3 the resonance frequency of Zb. The variations of the impedances 2a and Zb' with frequency are shown respectively by curves [5 and 16 of Fig. 10 and one form that the modified branches may take is illustrated by Fig. 11. The values of the branch elements may be determined from the resonance and antiresonance frequencies in the manner described by R. M. Foster in an article entitled A Reactance Theorem, Bell System Technical Journal, vol. III No. 2, April 1924.

v The application of the transformation of Figs. 1 and 8 to the modified prototype networks introduces additional impedance controlling factors while retaining those introduced by the modification of the prototype. The two processes, therefore, supplement each other and together provide for complete control of the image impedances.

The derivation processes described in the foregoing, may be applied to prototype networks of any type, but their greatest utility lies in their application to wave filter networks. It has already been pointed out that the most general transfer constant of a wave filter can be represented as the sum of the transfer constants of a symmetrical portion, which may be constructed as a symmetrical lattice network in accordance with my earlier Patent 1,828,454 issued October 20, 1931, and of a number of simple series-shunt type half sections. The image impedance of the symmetrical lattice may be made to have any possible characteristic by the appropriate allocation of the branch impedance resonances and anti-resonances, but in order that the completely general transfer constant may be obtained it is necessary that the additional half sections be capable of assuming impedances which match each other and that of the symmetrical lattice when connected in tandem. The complete control of the image impedances of unsymmetrical half sections by means of the processes described above permits the required impedance matching to be accomplishedand so makes possible the construction of wave filters of completely general characteristics.

An example of a composite filter employing the networks of the invention is shown in Fig. 12. This isa low-pass filter network comprising a symmetrical lattice F1; and two unsymmetrical networks F2 and F3 represented respectively by the portion of the network between dotted AA and BB and the portion between BB and CC. The network F2 is an h-derived network corresponding to the elementary prototype half section of Fig. 13, the term h-der'ived network being used herein to define a network related to a prototype in accordance with the principles of the invention as described above. The network F3 is i an h-derived network corresponding to the prototype half section shown in Fig. 14 which is an stant and the first image impedance are chosen are both alike are represented by curve I! of Fig. 15. The characteristic has one pole at frequency in above the cut-oif frequency f1, this providing a single impedance controlling factor. Theimage impedance at the other end of the network is represented by curve I8 of Fig. 16 and is characterizedby a pole at frequency f2 and a zero at a higher frequency is. The zero in this case corresponds to the frequency of the attenuation peak of the prototype half section of Fig. 14.

The difference inthe two image impedances of the composite filter arises from the fact that, in the case of reactive networks, if the transfer constant is given only one image impedance can be chosen arbitrarily, the second image impedance being determined as soon as the transfer con- What is claimed is:

1. The method of building out a series shunt transmission network comprising a series impedance Za and a shunt impedance Zb to produce a new network having the same transfer constant as the original network, but having different image impedances therefrom, which comprises alternate steps of adding series branches having impedances equal to (1h)/h times the short-circuit impedances of the networks to which the said branches are added, it being a positive numeric less than unity, and shunt branches having impedances equal to h/(l-h) times the open circuit impedances of the networks to which the said shunt branches are added, the Z impedances throughout the network at each stage being diminished in accordance with the factor 71. before a new series branch is added and the Z impedances being increased in accordance with the factor l/ h before a new shunt branch is added.

2. The networkaccording to claim 1 in which the value of the factor it is changed with each step of adding a newbranch.

3. A four-terminal network having the same transfer constant as a prototype network consisting of a single series impedance Za and a single shunt impedance Zb, comprising a series branch having an impedance klZa, a shunt branch having an impedance Zb/Icz and a built out portion comprising a plurality of successively added series and shunt branches each series branch having an impedance equal to times the short-circuit impedance of the portion of the network to which it is added and each shunt branch having an impedance equal to h'/ (l-h') the open-circuit impedance of the portion of the network to which it is added, the quantities h and It being positive numerics less than unity which have Values hi, hz, --hn, respectively for the successively added series branches and hi, h2'-hn ance proportional to the open-circuit impedance of the portion of the network to which it is added.

5. A four-terminal transmission network having the same transfer constant as a prototype network comprising a single series impedance Za and a single shunt impedance Zb, comprising a series branch having an impedance hZa, a shunt branch of impedance Zb and a second series branch having an impedance (1-h) /h times the impedance of said first mentioned series branch and said shunt branch in parallel, where h is a positive numeric less than unity.

6. A four-terminal transmission network having the same transfer constant as a prototype network comprising a single series impedance Z3, and a single shunt impedance Zb, comprising a shunt branch having an impedance Zb/h, a series branch havingan impedance Zaand asecond shunt branch having an impedance equal to h/(l-h) times the impedance of said first mentioned shunt branch and said series branch in series.

'7. A composite wave filter comprising a reactance network proportioned to transmit currents of frequencies in a pro-assigned band and to attenuate currents of other frequencies, and having a pre-assigned image impedance which has the same value at both the input and the output terminals of the network, and connected in tandem therewith, an h-derived network having the same transfer constant as a prototype half-section wave filter adapted to transmit the same band of frequencies as said first mentioned network, but having image impedances which differ from that of said first mentioned network, said h-derived network having an image impedance which matches that of said first mentioned network at the junction point.

8. A composite wave filter comprising a plurality of h-derived networks connected in tandem, said h-derived networks having the same transfer constants as respective prototype half-section wave filters comprising single series branches and single shunt branches, but having image impedances differing from those of the respective prototypes by the inclusion of additional resonance frequencies in the attenuating ranges, and the image impedances of said h-derived networks being further proportioned to provide impedance matching at the junction points of the several networks.

. HENDRIK W. BODE. 

